Deprecated: Caller from MathObject::readFromCache ignored an error originally raised from MathObject::readFromCache: [1054] Unknown column 'math_mathml' in 'field list' in /var/www/html/includes/debug/MWDebug.php on line 379
Deprecated: Caller from Wikibase\Lib\Store\Sql\SiteLinkTable::getItemIdForLink ignored an error originally raised from MathObject::readFromCache: [1054] Unknown column 'math_mathml' in 'field list' in /var/www/html/includes/debug/MWDebug.php on line 379 Modular lambda function - LaTeX CAS translator demo
Modular lambda function
From LaTeX CAS translator demo
Revision as of 01:00, 9 September 2019 by imported>InternetArchiveBot(Redirected page to wmf:Privacy policy)
In mathematics, the elliptic modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curveX(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve, where the map is defined as the quotient by the [−1] involution.
By symmetrizing the lambda function under the canonical action of the symmetric group S3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group , and it is in fact Klein's modular j-invariant.
which is the j-invariant of the elliptic curve of Legendre form
Little Picard theorem
The lambda function is used in the original proof of the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.[8] Suppose if possible that f is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function z → ω(f(z)). By the Monodromy theorem this is holomorphic and maps the complex plane C to the upper half plane. From this it is easy to construct a holomorphic function from C to the unit disc, which by Liouville's theorem must be constant.[9]